Investment Flexibility and the Acceptance of Risk

File: DISTIL 230001 . By:DS . Date:01:10:97 . Time:09:53 LOP8M. V8.0. Page 01:01Codes: 4102 Signs: 2234 . Length: 50 pic 3 pts, 212 mm

Journal of Economic Theory � ET2300

journal of economic theory 76, 219�241 (1997)

Investment Flexibility and the Acceptance of Risk*

Christian Gollier

GREMAQ and IDEI, Universite� de Toulouse, Place Anatole France,31042 Toulouse Cedex, France

and

John Lindsey and Richard J. Zeckhauser

Kennedy School of Government, Harvard University, Cambridge, Massachusetts 02138

Received May 6, 1996; revised February 26, 1997

The hypothesis examined in this paper is that the greater the investor's flexibility,the easier it is for him to change his portfolio depending on his results, the morewilling he will be to accept risks. When the investor has no control on the size ofthe risky investment, but can choose between one risky and one riskless asset, thisconjecture is shown to be correct. However, if there is more than one risky asseteach period, counterexamples demonstrate that flexibility rarely ensures greater risktaking. For the standard portfolio problem in which investors are free to determinethe size of their investment in a risky asset, flexibility always raises the demand forthe risky asset if constant relative risk aversion is less than unity. But counter-examples can always be found when the constant relative risk aversion is largerthan unity. Journal of Economic Literature Classification Numbers: G11, D81, D84.� 1997 Academic Press

1. INTRODUCTION

Common wisdom suggests that the greater one's ability to shift one'sinvestment after intermediary shocks to wealth, the riskier should be one'sportfolio. For example, an investor with decreasing risk aversion shouldbe more likely to take a risky investment if he can exit should returnsplummet, that is when the costs of risk escalate.

This analysis examines the relationship between flexibility��we label itsopposite rigidity��and the risk of portfolios optimized to maximize expectedutility. The strong form of rigidity arises when the investor is not able

article no. ET972300

2190022-0531�97 �25.00

Copyright � 1997 by Academic PressAll rights of reproduction in any form reserved.

* The authors are grateful to an anonymous referee for helpful comments. Research supportfrom TIAA-CREF is greatly appreciated. Miriam Avins and Mihir Desai provided excellentassistance.


to modify his exposure to risk from period to period. In this paper, weexamine a somewhat weaker form of rigidity, i.e., the investor can modifyhis risk exposure from one period to another, but he is forced to fix it apriori, i.e., before observing realizations of the early returns on his invest-ment. We simplify the problem by considering a two-period model. Theindividual invests at the beginning or period 1, receives investment returns,and then must invest in period 2. In the rigid economy, period 2 investmentdecisions must be made before period 1 results are learned.

In Section 2, we consider a model in which the investor can only investin indivisible investment projects. Our main result is that, except for ahandful utility functions, it is always possible to build a counterexample inwhich less risk will be taken in the flexible economy than in the rigid one.We exclude the case of constant absolute risk aversion, for which wealthhas no effect on risky investment choice, hence flexibility has no effect onthe risk of the optimal portfolio.

Section 3 is devoted to the standard portfolio problem. We assume thatone can invest in two assets, one being risk free. The returns of the riskyasset are not serially correlated. In this case, the two forms of rigiditiescoincide, since the investor who is forced to determine today the size of hisinvestment in the risky asset tomorrow will invest the same amount in therisky asset in each of the two periods. The impact of flexibility upon riskhas important consequences in the context of portfolio management. In theUnited States, for example, retirement funds are more flexible thanordinary investments because capital gains taxes need not be paid whenretirement investments are shifted from one security to another. Thissuggest a fundamental maxim due to flexibility: an individual with bothordinary savings and a retirement account should emphasize (riskier)assets in the latter.1

2. THE CASE OF INDIVISIBLE INVESTMENT PROJECTS

2.1. The Model

We consider a simple two-period model. The investor is endowed withwealth w at the beginning of the first period. He maximizes the expectedutility of his final wealth. The utility function u is assumed to be twice dif-ferentiable, increasing and concave. A(z)=&u"(z)�u$(z) denotes absoluterisk aversion. In each period t=1 or 2, he must choose one of twoindivisible investment projects. The net payoffs of the two projects available

220 GOLLIER, LINDSEY, AND ZECKHAUSER

1 This parallels the well known result that the tax-protected retirement funds should dis-proportionately be allocated higher current yield assets.


at period t are denoted respectively x~ t and y~ t . Random variables x~ 1 , x~ 2 , y~ 1 ,y~ 2 are independently distributed.

We compare two economies which differ on the timing of decisions. Inwhat we call the ``rigid'' economy, the investor has to decide at the begin-ning of the first period about which project to undertake at each period. Itis a ``rigid'' economy in the sense that the investor cannot reexamine hissecond period risk exposure due to first period results, perhaps large lossesor gains. The problem of the investor in the rigid economy is to find thepair ($r

1 , $r2) that solves the following program:

max

$2 # [0, 1]$1 # [0, 1]

Eu \w+ :2

t=1

($t y~ t+(1&$t) x~ t)+ . (1)

It is a simple maximization at the beginning of period 1. Choice variable$t takes value 1 if project y~ t is undertaken in period t, otherwise $t=0 andproject x~ t is undertaken in period t.

We compare this optimal risk-taking decision with the optimal deci-sion in the so-called ``flexible'' economy. In the flexible economy, theinvestor can delay the decision on which project to undertake in period 2until he has learned (observed) the return of his first period investment. Inthis economy, the dynamic problem of the investor is solved by using back-ward induction:

$ f1 # arg max

$1 # [0, 1]Ev(w+$1 y~ 1+(1&$1) x~ 1). (2)

Function v is the value function which is obtained by solving the secondperiod problem:

v(z)= max$2 # [0, 1]

Eu(z+$2 y~ 2+(1&$2) x~ 2). (3)

Some comments are in order here. First, observe that program (1) can beseen as a constrained version of the dynamic problem, represented by equa-tions (2) and (3), faced by the investor in the flexible economy. The con-straint in the rigid economy is that $2 must be independent of z, the wealthlevel at the beginning of the second period. The investor is strictly better-offin the flexible economy. There is one exception to this rule, that is when theoptimal decision of the investor in period 2 is independent of his wealth z.This will happen if x~ 2 and y~ 2 are identically distributed, if one set of gam-bles stochastically dominates another, or if all outcomes lie in a constantabsolute risk aversion (CARA) portion of u. In the CARA case, decisionsare not affected by wealth; hence learning period 1 results before decidingon period 2 investments offers no advantage.

221FLEXIBILITY AND RISK ACCEPTANCE


It is worth noting that the value function v used by the investor in theflexible economy to solve his first period need not be concave. This wasfirst noticed by Bell [1].2 The existence of a discrete choice in period 2, ashere, can induce a risk-loving behavior in period 1. As it is intuitively sen-sible, ``an entrepreneur with an idea she believes will work, but withoutfinancial backers who agree, would be justified trying to raise the necessarycapital in Atlantic City'' (Bell [1]).

2.2. A Positive Result when one of the Two Projects is Certain

We first consider a simple special case of the problem presented above.The relationship between flexibility and risk is most easily understoodwhen the investor must choose between a risky and a risk-free position. Weassume here that one of the two projects available in period 1 is risk-free.Without loss of generality, we assume that x~ 1 is zero with probability 1.The investor's problem in period 1 is just to accept or to reject a riskyinvestment project y~ 1 .

Common wisdom suggests that more flexibility in risk-taking decisionsinduces people to be more willing to accept risks. This logic would meanthat the first period investment is undertaken in the flexible economy if itis undertaken in the rigid economy. This proves to be true, as shown in thefollowing Proposition.

Proposition 1. If x~ 1 is zero with probability 1, then the risky project isalways undertaken in the flexible economy if it is undertaken in the rigid one:$r

1=1 O $ f1=1.

Intuition. Introducing flexibility is not about to make you switch to thesafe project in period 1, since with a safe project flexibility is uselessbecause you know what is going to happen in period 1.

Proof. If the risky project is not undertaken in period 1, the expectedutility of the investor is the same in the two economies, since waiting todecide is not worth anything when one knows what is going to happen inperiod 1. The expected utility in this case equals

U f ($1=0)=Ur($1=0)= max$2 # [0, 1]

Eu(w+$2 y~ 2+(1&$2) x~ 2). (4)

Also, because the information has a value, when the risky project isundertaken in period 1, the investor can obtain a larger expected utility inthe flexible economy than in the rigid one, i.e., U f ($1=1)�U r($1=1).Finally, assume that it is optimal to undertake the risky project in period 1


2 For additional insights on the concavity of the value function, see Pratt [9]. Gollier [5]analyzed the dynamic problem of the investor in this flexible economy when risks are i.i.d. over time.


in the rigid economy, i.e., Ur($1=1)>Ur($1=0). Combining these condi-tions yields

U f ($1=1)�Ur($1=1)>Ur($1=0)=U f ($1=0). K (5)

In this simple framework, the value of the information in the flexibleeconomy not only increases the expected utility of the investor, but it alsomakes him less risk-averse towards the first period project. This is not truein general, as the next section explains.

2.3. A General Counterexample: Flexibility Need Not Increase Risk Taking

Once the choice set is allowed to contain more than one risky projectfrom period to period, it is less likely that flexibility will promote risk tak-ing in all circumstances. In effect, we find that��except for utility functionsthat satisfy a very specialized, unintuitive condition��a counterexample willalways exist to the assertion that flexibility-promotes-risk taking.

This result is particularly surprising for utility functions displayingmonotonic risk aversion. With such utility functions, the riskier the first-period lottery, the greater will be the diversity in risk aversion (risks costs)depending on the first-period outcome. Hence, second-period investmentsthat are attractive for some first-period outcomes will be unattractive forothers. The more diverse is a utility function's relative risk aversion, itwould seem, the greater the benefits reaped from flexibility in the choice ofsecond-period lottery. Hence, we might conjecture that if the riskier first-stage lottery is optimal in the rigid economy, it would also be optimal inthe flexible economy as in Proposition 1. Our Proposition 2 below shows,however, how to construct a counterexample to this conjecture for quitegeneral utility functions, including rather general monotonic risk averseutility functions.

In the previous section, risk y~ 1 was obtained by adding some noise torisk x~ 1 which was degenerate at zero. Here we extend here this definitionby allowing x~ 1 to be any random variable.

Proposition 2. Suppose that there exists a scalar z such that u$$$$ is con-tinuous at z and A"(z){A$(z) A(z). Take y~ 1 by adding some noise =~ to somerealization of x~ 1 . Then, there exists x~ 1 , =~ , x~ 2 , y~ 2 such that the investorprefers the first period lottery y~ 1 in the rigid economy, but the lottery x~ 1 inthe flexible economy.

Proof. See the Appendix.

In the remainder of this section, we provide some intuition for this result.Let us assume that A"(z)>A$(z) A(z). The case A"(z)>A$(z) A(z) corre-sponds to the notion of strict ``local properness'', as introduced by Pratt



and Zeckhauser [10]. To illustrate, consider two lotteries with payoffs =~ 1and =~ 2 . These lotteries are small and their payoffs are independently dis-tributed. Suppose that if you were offered either lottery, you would beindifferent whether or not to accept it. Might you accept the two lotteriesif they were offered jointly? If your answer is no, as we might thinkintuitively to be appropriate, then your risk aversion is said to be strictlylocally proper. Small risks are not complementary: taking on one smallundesirable (or neutral) risk does not make another small undesirable(or neutral) risk desirable. It can be inferred from the work by Pratt andZeckhauser that risk aversion is strictly locally proper if A"(z)>A$(z) A(z).

Under strict local properness, one can build a counterexample asfollows: normalize w0 to zero, and consider x~ 1 which takes values z and zwith some probabilities. Lottery y~ 1 is obtained from x~ 1 by adding somenoise =~ . Assume that =~ is zero with probability 1 if x~ 1=z, and =~ is somesmall risk =~ z1 if x~ 1=z. We assume that Eu(z+=~ z1)=u(z), i.e., a myopicagent would be indifferent between x~ 1 and y~ 1 . In period 2, we have x~ 2

equal to zero, and y~ 2 is some small risk such that Eu(z+y~ 2)=u(z). Inshort, at each period a myopic investor with wealth z would be indifferentbetween the two projects.

In the flexible economy, the investor selects project y~ 2 in period 2 if hegot payoff z in period 1. Since we assumed that risk aversion is locallyproper at z, it means that noise =~ z1 which was considered to be neutral bythe myopic agent is strictly disliked by the dynamic investor in the flexibleeconomy. Since y~ 1 differs from x~ 1 only by the addition of noise =~ z1 if x~ 1=z,the option to invest in y~ 2 in period 2 makes the flexible investor prefer thesafer x~ 1 in period 1. This illustrates the fact that a small neutral risk��hererisk y~ 2��can substitute for any other small risk (=~ z1).

Since risk aversion is locally proper at z, A may not be maximal at z. So,select z such that A(z)>A(z). In the rigid economy, the investor evaluatesthe desirability of the second-period lottery y~ 2 without knowing whetherhis wealth in period 2 will be z, z, or z+=~ z1 . Recall that the investor isneutral to y~ 2 at z. Since =~ z1 is small and A is continuous, this is also almosttrue at z+=~ z1 . But as A(z)>A(z), the investor strictly dislikes y~ 2 at wealthz. Therefore, the investor in the rigid economy does not undertake the riskyproject in period 2. He behaves in period 1 like the myopic investor, i.e.,he is indifferent between x~ 1 and y~ 1 . A slight modification in the distributionof the noise can induce him to prefer the riskier project y~ 1 . This is aparadox since the same investor prefers the less risky project x~ 1 in theflexible economy.

This discussion suggest that paradoxes may arise when investment rulesare rigid, because risky positions may be transferred from one period toanother. In the general counterexamples we construct for the case A"(z)>A$(z) A(z), the investor in the rigid economy takes a riskier position in



period 1, but accepts no risk at all in period 2. In the flexible economy, aless risky project is taken initially, but the loss in expected return is morethan made up by sometimes taking a risky project in period 2.

Let us consider the following numerical example to illustrate this con-struction. Take u(t)=log(t), a locally proper utility function. Initial wealthis normalized to 1. Lottery x~ 1 is distributed as (z=0, 1�2; z=2, 1�2),whereas lottery y~ 1 is obtained from lottery x~ 1 by adding risk =~ z1=(1.5, 1�2;&1, 1�2) to the good realization of x~ 1 . In period 2, the decision-maker hasan option to invest in y~ 2=(2, 1�2; &1, 1�2) or not. In the flexible economy,it is easy to check that this option is used in period 2 if wealth at that timeis larger or equal to 2. In consequence, independent of whether or not theagent purchases risk =~ z1 or not, the option to invest is used in period 2 ifthe realization of x~ 1 is 2. Two observations lead to the counter-example.First, given wealth w0=1, the agent is indifferent between x~ 1 and y~ 1 whenhe has no option to take risk in the future. Second, it can be shown thatthe second-period option on y~ 2 increases local risk aversion with respectfirst-period risks in the interval [2, 4+3 - 2] of first-period wealth; v isstrictly more concave than u in this interval. Since y~ 1 is riskier than x~ 1 inthis interval, and is the same elsewhere, the flexible investor now prefers x~ 1

due to the existence of the option on y~ 2 . In short, the flexible investorselects the less risky x~ 1 in the first period, and he invests in y~ 2 in case ofa success. In contrast, the rigid investor never invests in y~ 2 , since he wouldotherwise finish with a zero wealth with some positive probability (1�4).Given this, he invests in y~ 1 , the riskier investment.

This example illustrates the fact that the existence of counter-intuitiveresults are not limited to the case of adding extremely small noises to theinitial gamble, nor to the case involving options on small risks in thefuture. The problem of larger risks is more difficult, as shown by Pratt andZeckhauser [10]. Even global properness is not very useful for ourproblem, since global properness is the condition for any undesirable(future) risk not to make any other undesirable risk desirable. Since weconsider options on future risks, only the effect of desirable risks matters.However, by continuity, if a neutral risk generates the result, it will alsopertain for risks that are not too much desirable. This is the case for risksy~ 2 for z not too much larger than 2, as in the example. Indeed, the accep-tance of y~ 2 increases local risk aversion for any wealth level between 2 and4+3 - 2, which is in fact ``much larger'' than 2.

In the case of A"(z)<A$(z) A(z), we know that a small neutral riskalways reduces the aversion to any other small risk: small risks are com-plementary. This counter-intuitive effect allows for a construction sym-metric to the one presented above. The flexible investor selects the saferasset in period 1 and sometimes selects the safer asset in period 2, while the



rigid investor commits to holding the riskier asset in both periods. Theintuition is that bearing more risk in period 2 in the rigid economy inducesthe investor to accept more risk in period 1 by complementarity. A fulldescription of how to build a counterexample in this case is presented inthe Appendix. Building a numerical example in this case is not simple, sinceall familiar utility functions we know satisfy the local properness condition.

Such constructions might not be possible when A"(z)=A$(z) A(z), i.e., ifthe absolute risk aversion is not equal to minus the absolute risk aversionof the absolute risk aversion. In this case, accepting a gamble on a smallrisk for which one is indifferent does not affect the willingness to gamble onanother small risk. For such utility functions, one might have a 5th-orderpreference. Except for the CARA class, &ez, z, the set of utility functionsthat satisfy A"=A$A is rather exotic: It contains u(z)=z3, u(z)=z,u(z)=z&sin(z), u(z)=z&sinh(z), u(z)=z+sinh(z), u(z)=&e&Az, andall u obtained by replacing u(z) by a+bu(c+dz).

3. THE PORTFOLIO PROBLEM WITH CONSTANT RELATIVERISK AVERSION

In this section, we examine a two-period portfolio problem with oneriskfree asset offering zero return and one risky asset with returns y~ 1 andy~ 2 respectively in period 1 and 2. Moreover, we assume that y~ 1 and y~ 2 arei.i.d. and distributed as y~ . The expectation of y~ is positive. As before, weexamine the effect of flexibility on the optimal exposure to risk in period 1.

In the rigid economy, the investor must determine the structure of hisportfolio at each period at the beginning of the first period. The problemof the investor in the rigid economy is written as3

(:r1 , :r

2) # arg max:1 , :2

Eu \w+ :2

t=1

:t y~ t+ . (6)

An important difference from problem (1) is that :t is not restricted to bezero or one. The size of the exposure to risk is now a decision variable.A simple diversification argument4 implies that the solution to problem (6)must be symmetric. Suppose by contradiction that it is not. Then, we have

:1 y~ 1+:2 y~ 2=:1+:2

2( y~ 1+y~ 2)+

:1&:2

2( y~ 1&y~ 2), (7)


3 Eeckhoudt, Gollier and Levasseur [4] compares :r1 with the optimal solution for the one-

period problem max: Eu(w+:y~ ).4 See for example Rothschild and Stiglitz [11].


where E[ y~ 1&y~ 2 | y~ 1+y~ 2=z] is obviously zero for any z. This means that:1 y~ 1+:2 y~ 2 is a Rothschild�Stiglitz increase in risk of ((:1+:2)�2)(y~ 1+y~ 2)in comparison to a symmetric choice. Therefore, the optimal solution to (6)must be symmetric. The problem in the rigid economy is thus equivalent to

:r # arg max:

Eu(w+:( y~ 1+y~ 2)). (8)

As before, we compare this optimal risk-taking decision with the optimaldecision in the flexible economy. In such an economy, the dynamicproblem of the investor is solved by using backward induction:

: f1 # arg max

:1

Ev(w+:1 y~ 1). (9)

Function v is the value function which is obtained by solving the secondperiod problem:

v(z)=max:2

Eu(z+:2 y~ 2). (10)

As is well-known in dynamic finance, the problem of the investor in theflexible economy is substantially simplified if we assume constant absoluterisk aversion (CARA), or constant relative risk aversion (CRRA)5, 6. Theproblem is trivial under CARA, since we obviously get :r

1=: f1 . Under

CRRA, myopia is optimal in the flexible economy. Myopia means that theinvestor considers each period in isolation to determine his portfolio. Thisis due to the fact that v inherits the same degree of concavity as u when uis CRRA. Therefore, we hereafter consider the case of u$(z)=z&#, #>0.Since u and v are equivalent, the first-order condition for program (9) canbe written as

E[ y~ 1u$(w+: f1 y~ 1)]=0. (11)

We are interested in determining whether the demand for the risky assetin the rigid economy is less than the demand for the same asset in theflexible economy, i.e., whether :r is less than : f

1 . Since the objective func-tion in (8) is concave in the decision variable, this is true if and only if

E( y~ 1+y~ 2) u$(w+: f1 ( y~ 1+y~ 2))�0. (12)


5 Gollier and Zeckhauser [7] consider other utility functions. They obtained the necessaryand sufficient condition on the utility function to assure that : f

1 is larger than the optimaldemand for the risky asset in the one-period�myopic problem, max: Eu(w+:y~ 1 ).

6 Caballe� and Pomansky [2] examined the properties of utility functions exhibiting mixedrisk aversion, as is the case for CRRA functions. They discuss the implication of mixed riskaversion on portfolio selection, and on the management of multiple risk taking.


We begin with a positive result which holds for small risks.

Proposition 3. Let u$(z)=z&#, #>0. If y~ is small in the sense that| y~ Ey~ �Ey~ 2 | is small almost surely, how small depending on #, then the demandfor the risky asset is smaller in the rigid economy than in the flexible.

Proof. See the Appendix.

The outline of the proof goes as follows: suppose that : f1 y~ 1 is small

almost surely. Then condition (11) can be approximated by

Ey~ 1 [u$(w)+u"(w) : f1 y~ 1]$0, (13)

or : f1=(1�A(w))(Ey~ �Ey~ 2). Thus our assumption above holds by assuming

that y~ (Ey~ �Ey~ 2) is small. We may thus also approximate u$(w+: fs ( y~ 1+y~ 2))

in (12) by a second order Taylor approximation around w. It yields

E( y~ 1+y~ 2)[u$(w)+u"(w) : f1 ( y~ 1+y~ 2)]�0. (14)

This is equivalent to

2u$(w) Ey~ +2u"(w) : f1 [Ey~ 2+(Ey~ )2]�0. (15)

By eliminating : f1 , inequality (15) is equivalent to &2u$(w)((Ey~ )3�Ey~ 2).

This is negative.This result does not hold for larger risks. As a counterexample, consider

the CRRA utility function with relative risk aversion #=5. Take y~ =(&0.1,4�10; 10, 6�10). After some computations, we get :r

1=0.1707, which islarger than : f

1=0.1678. Our next Proposition provides a necessary and suf-ficient condition for : f

1�:r1 under CRRA.

Proposition 4. A necessary and sufficient condition for : f1�:r

1 is thatthe constant relative risk aversion be less than or equal to unity. It isnecessary in the sense that if # is larger than unity, than one can always findan initial wealth z and a distribution y~ such that : f

1 if less than :r1 .

Proof. See the Appendix for the proof of the general case. We provehere that : f

1�:r1 for the logarithmic (#=1) investor, since this case does

not require using sophisticated techniques. We have to prove that for anyw and any y~ 1 , y~ 2 ,

Ey~ i

w+: f1 y~ i

=0 O Ey~ 1+y~ 2

w+: f1 ( y~ 1+y~ 2)

�0. (16)



Notice that, for all z, w+z, z�(w+z)=w((1�w)&(1�w+z)). Also, becauseEy~ >0 by assumption, : f

1 is positive. Thus, the right condition in (16) isequivalent to

E _1w

&1

w+: f1 ( y~ 1+y~ 2)&�0. (17)

The left condition in (16) can be rewritten in a similar way, implyingthat it is equivalent to

Eh(w+: f1 y~ i)=h(w) O Eh(w+: f

1 y~ 1+: f1 y~ 2)�h(w). (18)

where h(z)=&1�z. Observe that h can be seen as a CRRA utility function.Condition (18) is interpreted as whether two lotteries : f

1 y~ 1 and : f1 y~ 2 for

which individual h is indifferent when considered in isolation, are jointlyweakly undesirable. We know that this is true, since function h is proper(see Pratt and Zeckhauser [10]). Though successful for the logarithmiccase, we were unable to extend this trick to other CRRA functions. K

4. THE CONCEPT OF FLEXIBILITY PREMIUM

In this section, we propose a way to quantify the effect of rigidity onwelfare. We consider an economy with several assets and potentially severalperiods, with either flexibility or rigidity in the decision process. We com-pare this economy with a ``dual'' economy with only one asset, which isrisk-free. The Certainty Equivalent Interest Rate (CEIR) is the per-periodrate of return of the riskfree asset in the dual economy which makes theinvestor indifferent between living in the reference economy or in its dualone.

Let us consider the case of an economy with a single period and twoassets: one is riskfree with a zero return, and the other is risky with return&100 or +200 with equal probability. Investor's utility is logarithmic.After some computations, it appears that his optimal strategy is to invest2500 of his wealth in the risky asset. Normalizing wealth to unity,7 themaximum expected utility of the investor equals 0.0588. The CEIR in thiscase is 6.070, since 0.0588=log(1+0.0607). The investor is indifferentbetween being an expected-utility-maximizer in the two-asset economy andliving in the dual economy with a risk-free rate of 6.070.

Consider now two economies, f and r. The investor is more constrainedin economy r than in economy f. We define the ``constraint premium'' as


7 In general, the CEIR will be a function of the investor's wealth. This is not case here, sincerelative risk aversion is constant.


the difference between the certainty equivalent interest rate of economy fand the certainty equivalent interest rate of economy r. It is necessarilynonnegative. It measures the loss in welfare due to adding the constraint inthe economy. The flexibility premium is an example of this, in which theconstraint is on the timing of the decisions.

Consider now the example above with 2 periods. In economy f, theCEIR is still 6.070. This results from the constancy of relative risk aver-sion, which makes myopia rational. In economy r, one can verify that theinvestor invests only 2300 of his wealth in the risky position, yielding aCEIR of 5.790. The flexibility premium is thus 0.280 per year. For three,four and five years, we get flexibilty premia of respectively 0.560, 0.850and 1.140 per year. In the case of the ``square root'' utility function, weget flexibility premia of respectively 1.830, 3.30, 4.50 and 5.650 peryear, for respectively two, three, four and five years horizons.

Technically, in the case of CRRA utility functions, the CEIR in theflexible economy is obtained by solving the static problem of the investor,since myopia is optimal,

u(1+CEIRf )=max:

Eu(1+R+:x~ ), (19)

where R is the risk-free rate and x~ is the excess return of the risky asset.The computation of the CEIR in the flexible economy with utility functionswhich do not exhibit linear risk tolerance would be much more difficult.The CEIR in the rigid economy with n periods is implicitly defined by

u((1+CEIRr(n))n)=max:

u(1+R+:(x~ 1+x~ 2+ } } } +x~ n)), (20)

with x~ 1 ,..., x~ n independent and identically distributed as x~ . Finally, theflexible premium in an economy with n periods of investment equalsCEIRf&CEIRr(n).

5. PARALLEL PORTFOLIOS

The analysis thus far has focused on properties of and comparisonsbetween pure portfolios, all rigid or all flexible. Many real world investors,however, have both rigid and flexible investments. They have what we label``parallel portfolios.'' Given parallel portfolios, two questions come to mind.First, when will the flexible portfolio be relatively large enough that theindividual can achieve the same outcome he would were he totally flexible?Second, is the maxim we mentioned at the outset satisfied: will the optimalholdings in the flexible portfolio be at least as risky as those in the rigidone?



We investigate these questions in a multi-period context assuming thereis a safe asset, s, yielding a zero return, and a gambling asset, g, whosereturn can range from complete loss to arbitrarily large gains. We say theamount placed in the gambling asset is invested. You are not permitted tosell either asset short. (If you were, the fully flexible outcome could betrivially achieved.)

Let the net payoff for investing t be tx where x>&1 (you cannot losemore than your entire investment) and x has density f (x). Let u be theutility function and w the initial wealth. Then the expected utility isr(t)=��

&1 u(w+tx) f (x) dx. As is well-known, the optimal t has the samesign as Ex~ . Also, the demand for the gambling asset is proportional towealth in the CRRA case, whereas the demand is independent of wealth inthe CARA case. Finally, it can easily be shown that, for any amount d ofthe gambling asset, one can find distributions of returns of the gamblingasset with an arbitrarily large return in some states for which d is theoptimal demand, in the case of CRRA, or in the case of CARA.

Suppose that we have several periods of investment, and a flexibleportfolio, f, where we can trade after each period and a rigid portfolio, r,whose holdings must be maintained. Let bij be the amount placed inportfolio i in the j th asset. Thus bfg is the amount placed in the gamblingasset in the flexible portfolio, and bg=bfg+brg would be the total invest-ment in the gambling asset. We shall investigate the nature of and potentialfor optimal solutions with the CARA utility function u(x)=&e&kx and theCRRA utility function u(x)=x1&#�(1&#), #>0.

5.1. CARA Utility

In the CARA case, the optimal level of g is independent of wealth. If theinvestor places some of his monies in the gambling asset of his rigidportfolio, he faces the risk of a large increase in the value of this asset,thereby overinvesting in it in the future. Also, since one cannot be short onthe safe asset, the investor should invest enough in the flexible portfolio inorder to have enough wealth to invest in the gambling asset at each period,even in the worst case of repeated losses. We thus obtain the followingsimple result.

Proposition 5. You can ensure investing exactly t each period for nperiods iff the initial bf �nt.

Proof. You cannot invest anything in the rigid portfolio since it mightrise above t forcing you to be overinvested in the next period. Thus wemust invest t each period in the flexible portfolio. The worst-case scenario(approachable anyway) is that the gambling investment goes to 0 eachperiod. Then you need bf�nt. K



5.2. CRRA Utility

In the CRRA case, the optimum ratio of monies allocated to the gam-bling and safe assets, a ratio we label a, is independent of wealth and time-horizon. The following lemmas and proposition relate to a utility functionwith CRRA.

In this section through Proposition 6 both assets could have been gam-bling and they need not be independent of each other. All that is neededis that each can greatly outperform the other sometimes (that is if value ofone is multiplied by r and the other by s in a period, then r�s can bearbitrarily close to zero or infinity.). Even without this last condition thesufficient condition for being able to invest as well as we could have withcomplete flexibility still holds.

Lemma 1. Suppose the initial brg is forced to be 0. Let fn=(1+a)n&1.One will be able to allocate a times as much to the gambling asset as safeasset for n periods iff bf � fnbr .

Proof. We use induction on n, so suppose the lemma holds for n. Thefirst period portfolio must satisfy bfg=a(bfs+br). By the inductionhypothesis, we also need that what is left of bf after the first period mustbe at least fnbr . The worst case scenario is that the gambling investmentgoes to 0 so we need bfr�fnbr . Multiplying this by a+1 and adding tobfg=a(bfs+br) yields bf�[(a+1) fn+a] br=fn+1br . K

Lemma 2. Suppose the initial brs is forced to be 0. Let gn=(1+1�a)n&1.Then one can assure allocation exactly a times as much in the gambling assetas in the safe for n periods iff bf�gn br .

Proof. The gambling investment appreciating by a factor of z is equiv-alent to the safe investment being multiplied by a factor of 1�z. Thus thereis a symmetry beween safe and gambling investments that changes a for fn

to 1�a for gn . K

Lemma 3. One can ensure investing exactly a times as much in the safe forn periods iff for the first period bg=abs , bfg�gn&1brg , and bfs� fn&1 brs .

The middle inequality is forced by Lemma 2 when the safe asset dropsto zero and the last is forced by Lemma 1 when the gambling investmentdrops to 0. Assume all three comparisons. Suppose that gambling invest-ment goes up. Without any adjustment you would be overinvested in thesecond period. Ignoring the safe portions, if you adjusted the gambling por-tion of your portfolio so that it had the right fraction invested and bfg2

becomes b*fg2 by the proof of Lemma 2, b*fg2�gn&2brg2 . However, now youwould be underinvested because of the ``ignored'' safe portions. Thus you



do not need to trade away this much gambling stock and this lastinequality still holds. Then ensurance follows from an induction hypothesis.

Suppose the gambling investment goes down. This is equivalent to thesafe portion going up and the symmetric argument applies.

Proposition 6. One can assure investing exactly a times as much in thegambling investment as in the safe for n periods iff for the first period

bf�_(1+a)n

1+an &1& br . (21)

Proof. Suppose some b**

assignments do assure the required outcome.Then multiplying the equations of Lemma 3 by gn&1& fn&1, (1+a)(1+fn&1),and (1+a)(1+gn&1), respectively yields the inequality of the proposition.Conversely, suppose that proposition inequality holds. By treating some ofthe flexible portfolio as rigid we may assume equality. Letting

bfg=agn&1 �(1+gn&1), brg=a�(1+gn&1),

bfs=fn&1 �(1+fn&1), brs=1�(1+fn&1)

gives three equalities in Lemma 3 and therefore the following inequality,which is a linear combination:

(1+gn&1+a+afn&1) bf�( fn&1+fn&1gn&1+afn&1 gn&1+agn&1) br .

Using the defintions of fn&1 and gn&1 , this is rewritten as (21). Multiplyingthese figures by the appropriate constant gives the correct bf and br andensurance follows from Lemma 3. This completes the proof. K

When n=2 this becomes bf�[2a�(1+a2)] br so bf�br suffices. Supposebf=br . If a>1, the last inequality of Lemma 3 forces the flexible porfolioto be more safely invested than the rigid. If a<1, the middle inequality ofLemma 3 forces the flexible portfolio to be more riskily invested than therigid. In sum, this shows that depending on conditions, the optimal rigidportfolio can be riskier (have a higher proportion in the gambling asset) orless risky than the optimal flexible portfolio. Our conjectured maxim fails.

5.3. Cases Close to CRRA

We may also be interested in the behavior of portfolios when utilityis not pure CRRA. Such utilities are easily produced by mixing a smallcomponent of CARA with CRRA. Thus we consider the utility functionu(x)=log(x)&de&kx, for small (absolute) values of d. With d positive(negative), the fraction of the portfolio invested in the gambling asset tendsto decline (increase) with wealth.



Suppose the inequality of Proposition 6 is satisfied strictly. Then we maysplit off an auxiliary portion of the flexible portfolio so that the rest of itgives equality. Suppose the auxiliary is zw where w is the total wealth. Letp=z min(t, 1&t) where t=a�(1+a).

With u(x)=log(x), r(ws)=log w+S(s) where S(s)=��&1 log(1+sx)

f (x) dx is maximized at s=t. Let |d | be less than min(0.5, S(t)&S(t+p),S(t)&S(t&p)). Let v be the optimal investment for u(x)=&e&kx. Letu(x)=log x&de&kx. Since d�&0.5, u is concave. For all wealths theoptimal investment fraction must be in I=[t&p, t+p], since otherwiselog loses more that |d | from taking the fraction t, and the second summandof u cannot make this up.

Let there be two periods and start by investing t of the auxiliary andfollow the scheme of the Proposition 6 for the rest of your wealth. Scale theoriginal wealth so that v is the total investment. By Proposition 6, after thefirst period we can adjust the non-auxiliary part of our wealth so that weare fraction t invested. Since we invested the same way in the auxiliary asthe rest of our wealth the auxiliary is still fraction z of our wealth. If weinvest t of the auxiliary in the gambling investment we would be fractiont invested, with the proposition adjustment on the rest. We can adjust theauxiliary fraction by at least min(t, 1&t) either way and this adjusts thefraction of the total wealth fraction by at least p either way and this allowsus to make it the optimum fraction which lies in I.

In the following we can get by (do as well as with wealth flexible) withequality in Proposition 6, implying that no auxiliary portfolio is available.Suppose t=0.5 for the optimal investment fraction for u(x)=log x. Letu(x)=log x&de&kx with &0.5<d<0. As above take |d | sufficiently smallthat the optimal investment fraction must be in [0.25, 0.75] and let v bethe optimal investment in the gambling asset for u2(x)=&e&kx. Let bf =br=v and invest half of each in the gambling asset in the first of two periods.Suppose the gambling investment goes up in the first period leaving us withwealth at least 2v. By concavity r$2(0.5x)�r$2(v)=0 and r$(0.5w)�0. Thusour optimal second period investment lies in [0.5w, 0.75w]. Since brs is atmost 0.25w we can adjust bg to investing at least 0.75w. Thus we can investoptimally in the second period. The case that the gambling investment goesdown in the first period is similar. Conditions for more than two periodswill, of course, be more difficult to satisfy.

6. CONCLUSION

Economists have long recognized that risky investments drive theeconomy, and they have praised those who take the risks. Much of modernfinance, both in theory and in the marketplace, seeks to facilitate investors'



ability to diversify their portfolios and make them more liquid, henceflexible. Though finance theory has constructed beautiful models to addressdiversification, relatively little attention has been paid to flexibility.

The hypothesis examined here is that the greater the investor's flexibility,the easier it is for him to change his portfolio depending on his results, themore willing he will be to accept risks. When the investor has no controlon the size of the risky investment, but can choose between one risky andone riskless asset, this conjecture proves correct. However, if there is morethan one risky asset each period, as would be the case in the real world,counterexamples demonstrate that flexibility rarely ensures greater risktaking. For the standard portfolio problem, counterexamples can even befound when the constant relative risk aversion is larger than unity.

Our focus here has been on the behavior of the individual decisionmaker; however, society, which sets the rules, may have a concern for theaggregate results of individuals' decisions in the macroeconomics of finan-cial decision making. For example, in evaluating the tax treatment of retire-ment funds, policy makers look at the effects on aggregate savings rate.Presumably they are also concerned about effects on risk taking, theprimary subject of this paper, and financial market stability. Our finding ofa quite general counterexample to the maxim that flexibility promotes risktaking suggests that seemingly minor financial market constraints, or theirrelaxation, may have unexpected effects on securities prices. Moreover, ourexperience with sunspot equilibria (Shell [12], Jackson and Peck [8]) andrelated phenomena suggest that these effects may prove both substantialand unpredictable.

Factors that promote investment rigidity, such as trading costs, taxation,and exit charges, can be thought of as fulfilling the opposite function ofinnovative securities. In fact, many new derivative securities are designed toovercome rigidities, for example to hedge an appreciated position withoutrequiring a disposition that would incur taxes. (See Duffie and Rahi [3]for a useful review on financial market innovations.) Our analysis suggeststhat in theory at least, new securities will have an unpredictable effect onrisk taking.

Many developing nations limit capital ouflows, thereby reducing theflexibility of investments, and hence their attractiveness, but possibly damp-ening the type of speculative activity that often plagues securities marketsof such nations.8 We have shown how the microeconomics of risk taking


8 Chile prevents foreign investors in its securities markets withdrawing their money in lessthan one year. Eduardo Engel observes that this rigidifying factor may have saved his nationfrom a financial collapse that Mexico recently suffered. Woo-Chan Kim, South KoreanMinistry of Finance, tells us that his nation is concerned about opening its securities marketsfurther to foreign investors because they may respond excessively to events between Northand South Korea. (Personal communications, February 1996).


depends upon the flexibility of investments. Its macroeconomics, addressingwhich investments society undertakes and how they perform, will be no lessdependent.

APPENDIX

Proof of Proposition 2. On a small enough open interval containing z,u$, A"&A$A have constant signs and u" is continuous. Restrict everythingto this interval. Let A(z){A(z). Let :=u$(z)(A(z) A$(z)&A"(z)), ;=u$(z)(A(z)&(A(z)). Let B be the gamble paying h, 0.5(1+bh) of the time,otherwise, &h. Let D be the gamble paying h, 0.5(1+dh) of the time,otherwise &h. Let E denote expected utility with initial wealth z. Definef(h)=E(B+D)&E(B)&E(D)+E(0). Then f (0)=f $(0)=f "(0)=f $$$(0)=0and by Taylor's Theorem f (h)=f $$$$(k) h4�24 for some 0<k<h. Nowf $$$$(0)=6u$$$$(z)+12(b+d ) u$$$(z)+24bdu"(z) and when we substitutec=0.5A(z) for b and d, this becomes 6:. Since f $$$$(k) is continuous ink, b, d at k=0, b=d=c, there exists =>0 such that f $$$$(k) will have thesame sign as : if h, b, d are within = of 0, c, c respectively.

Let g(h)=E(B)&E(0)=0.5(1+bh) u(z+h)+0.5(1&bh) u(z&h)&u(z).Then g(0)=g$(0)=0 and by Taylor's Theorem g(h)=0.5g"(k) h2 for some0<k<h. Since u is strictly monotone, we can and do take b so thatg(h)=0 and then 0=g"(k)=0.5(1+bk) u"(z+k)+bu$(z+k)+0.5(1&bk)u"(z&k)+bu$(z&k). Solving for b:

b=&0.5(u"(z+k)+u"(z&k))

u$(z+k)+u$(z&k)+0.5k(u"(z+k)&u"(z&k)).

As h and k approach 0, b approaches c=0.5A(z). Take h small enough thatb is withing = of c. Take d between 0.5A(z) and 0.5A(z) and within = of c.Then we can and do take h small enough that E(B+D)&E(D)=f (h) hasthe same sign as :.

With initial wealth w=z or w=z\h, E(D)&E(0)=0.5g"(k) h2 forsome 0<k<h where

g"(k)=0.5(1+dk) u"(w+k)+du$(w+k)+0.5(1&dk) u"(w&k)

+du$(w&k).

As h approaches 0, g"(k) approaches u"(w)+2du$(w)=2(d&0.5A(w)) u$(w).Taking h small enough, g"(k) has the same sign as (d&c)u$, 0.5(A(z)&A(z))u$,and ;. When w=z, d&0.5A(w) reverses sign and g"(k) has the same signas &; for small enough h. Here g and k change as the base wealth changes,but we make h small enough to guarantee that all k's are sufficiently small.



To summarize, D is strictly desirable at z when &; is positive and strictlyundesirable otherwise, the opposite at z, z\h.

Adjust b slightly enough that E(B+D)&E(D) retains the same sign as: but E(B)&E(0) takes on the same sign as &:. Define the first periodlottery x~ 1 to have outcome z with probability p and z with probabilityq=1&p. Let the other first period lottery y~ 1 be the same, except we throwin the lottery =~ =B when z arises. Let the second period lotteries x~ 2 and y~ 2

be 0 and D, respectively. Take p large enough that in the inflexible worldwe chose D exactly when D is desirable at z. Define ES(X ) to be theexpected utility after the second period when we start with X in the firstperiod and, given this start and possible flexibility constraints, choosethe second period optimally. Since z always comes up with the same prob-ability and we always choose the same second period lottery when it doescome up, we can ignore the z-contribution to ES( y~ 1)&ES(x~ 1). When weuse D for the second period lottery at z, z\h, which is optimal given thefirst period lottery when &;>0 and the economy is inflexible, or &;<0and the world is flexible, ES( y~ 1)&ES(x~ 1)=q(E(B+D)&E(D)) which hasthe same sign as :. When we use 0 for the second period lottery at z, z\h,which is optimal given whichever the first period lottery when &;>0and the economy is flexible or &;<0 and the world is inflexible,ES( y~ 1)&ES(x~ 1)=q(E(B)&E(0)) which has the same sign as &:. Thus inthe flexible (resp. inflexible) world the sign of ES( y~ 1)&ES(x~ 1) is the sameas the sign of :; (resp. &:;). Thus by choosing A(z) on the sign (&:u$(z))side of A(z), x~ 1 is strictly optimal in the flexible world and y~ 1 is strictlyoptimal in the inflexible world. We can find such a z else A has the inap-propriate maximum or minimum at z and A$(z)=0. But then A(z)&A(z)has the same sign as A"(z) and &:u$(z) for z in some deleted neighbor-hood of z. K

Alternatively, suppose A"(z)=A$(z) A(z), A$(z)>0, and u$$$$ is continuousat z. Then the example still works: Let z be slightly larger than z so thatA(z)&A(z) has the same sign as A$(z). Let B be neutral at z and D at z+h.By the formula for b it is 0.5A(z) within second order terms in h. Similarly,d=0.5A(z+h)=0.5A(z)+0.5hA$(z) within second order terms. Sincef (h)=E(B+D)&E(B)&E(D)+E(0) is even in h we may use Taylor'sformula with just the term f $$$$(z)h4�24 and a sixth order error term. Then forsmall h the 0.5hA$(z) part of d dominates and is multiplied by 12u$$$(z)+24bu"(z) r 12u$$$(z) + 24(&0.5u"(z)�u$(z)) u"(z) =&12u$(z) A$(z) withinsecond order terms. Thus f (h) has the same sign 0.5hA$(z)(&12u$(z) A$(z))and &u$(z) and the example works.

Now suppose u$$$$ is continuous on an interval I. Assume A"&A$A iszero on I since otherwise the theorem guarantees an example. Solving thisfor A and then solving the definition for A for u, the solutions for u, within



the affine equivalence of u to a+bu(c+dz), are ez, z, z3, z&sin z,z&sinh z, and z+sinh z. The first two are CARA and there is no examplesince the preference between 0 and D is independent of wealth. The nextthree satisfy A$>0 and the last paragraph produces an example. Only forthe last u do we not know whether an example exists.

Proof of Proposition 3. Since u is CRRA, we may assume w=1. Let usalso assume without loss of generality that the supremum of | y~ | equals 1.Let

:$=1

A(1)EyEy2=

1#

EyEy2

and 0<|c&1|<1. Then by the definition of u", |c:$y|�2:$ can beassumed small enough that &| yc:$y(c&1) #�c|�yu$(1+c:$y)&yu$(1)&yc:$yu"(1)�| yc:$y(c&1) #�c|. Taking expectations in this inequality yields

&#:$ |c&1| Ey2�Eyu$(1+c:$y)&Ey+c:$#Ey2

=Eyu$(1+c:$y)+(c&1) :$#Ey2

�#:$ |c&1| Ey2.

When c>1 (c<1) the right-hand (left-hand) inequality showsEyu$(1+c:$y)�0 (�0) and :�c:$ (:�c:$). This shows we may make :relatively close (ratio close to 1) to :$ by making Ey�Ey2 small. In par-ticular assume 0.5:$<:<2:$<0.1.

The Taylor series for u$ about 1 converges uniformly on any symmetricclosed interval about 1 not containing 0. As |:( y1+y2)|�2:$( | y1 |+| y2 | )�(0.1)2=0.2 we may use this series on the LHS's of equations (11) and (12).Then the LHS of equations (12) minus twice equation (11) becomes

:�

k=1

1k!

u (k+1)(1) :k[E( y1+y2)k+1&Eyk+11 &Eyk+1

2 ]

= :�

k=1

1k!

u (k+1)(1) :k :k

i=1\k+1

i + EyiEyk+1&i

Now Ey<Ey2 by assumption and for j>1, |Ey j |�E | y2y j&2 |�Ey21 j&2=Ey2. Thus for the tail of our series



} :�

k=4

1k!

uk+1(1) :k :k

i=1\k+1

i + EyiEk+1&i }� :

�

k=4} 1k!

u(k+1)(1) :k2k+1(Ey2)2 }�

25:4

(0.5)4 :�

k=4

1k!

u(k+1)(1)(&0.5)k (Ey2)2

�29:4u$(1&0.5)(Ey2)2

=2#+9:4(Ey2)2.

The remaining terms are

u"(1) :2(Ey)2+ 12 u$$$(1) :26EyEy2+ 1

6u$$$$(1) :3[6(Ey2)2+8EyEy3]

=&#:[2(Ey)2&3(#+1) :EyEy2+(#+1)(#+2) :2(Ey2)2]

&4#(#+1)(#+2) :3EyEy3�3.

For the last term |:3EyEy3|�:3 |Ey| |Ey2|=:3:$#(Ey2)2�2#:4(Ey2)2. Ifwe substitute :$ for : in the bracket we get

2(Ey)2&3(#+1) :$EyEy2+(#+1)(#+2) :$2(Ey2)2

=[2#2&3(#+1) #+(#+1)(#+2)] :$2(Ey2)2=2:$2(Ey2)2.

This substitution introduces small relative error in each term in the bracewhere it is small absolute error which may be made small enough thatbefore the substitution the brace is at least 1 and the original bracket(before we substituted for :) is at least (:2�4)(Ey2)2. Then, with : suf-ficiently small &#: times the bracket is still negative even after adding inthe last term and the tail of the series. K

Proof of Proposition 4.

Without loss of generality, we hereafter normalize : f1 to be equal to

unity. We look at conditions on the utility function which guarantee thatthe following condition holds:

Ey~ 1u$(w+y~ 1)=0 and Ey~ 2u$(w+y~ 2)=0

O E( y~ 1+y~ 2) u$(w+y~ 1+y~ 2)�0, (22)

for any w, y~ 1 and y~ 2 , and for u$(z)=z&#. To solve this problem, we use the``bivariate diffidence theorem'' obtained by Gollier and Kimball [6]:



Theorem 1. Suppose that function h(x, y) satisfies h(x, 0)#0 andh(0, y )#0. Then, as long as f $1(0){0 and f $2(0){0, the two following condi-tions are equivalent:

v \y~ 1 , y~ 2 : Ef1( y~ 1)=f1(0) and Ef2( y~ 2)=f2(0) O Eh( y~ 1 , y~ 2)�0.

v h( y1 , y2) & $1( y1)( �h�y1

)(0, y2) & $2( y2)( �h�y2

)( y1 , 0) + $1( y1) $2( y2)( �2h

�y1 �y2)(0, 0)�0, for all y1 in the domain of f1 , y2 in the domain of f2 and

y1+y2 in the domain of h, with $i (t)=( fi (t)&fi (0))�f $i (0).

We apply this Theorem with h( y1W , y2) = ( y1 + y2) u$(w + y1 + y2) &y1 u$(w+y1)&y2u$(w+y2), f1( y1)=y1 u$(w+y1) and f2( y2)=y2u$(w+y2).Accordingly, condition (22) is equivalent to

H(#, w, y1 , y2)=( y1+y2) _u$(w) u$(w+y1+y2)u$(w+y1) u$(w+y2)

&1&&y1 y2[2A(w)&A(w+y1)&A(w+y2)]�0, (23)

for all w, y1 , and y2 such that w > 0, w + y1 > 0, w + y2 > 0 andw+y1+y2>0. Notice that H#0 under CARA. With u$(z)=z&#, condi-tion (23) can be rewritten as

H(#, w, y1 , y2)=( y1+y2) _(w+y1)# (w+y2)#

w#(w+y1+y2)# &1&&

#y1 y2[w( y1+y2)+2y1 y2]w(w+y1)(w+y2)

�0. (24)

The analysis of function H shows that it is nonpositive in the relevantdomain if # is less or equal to 1. Let us denote

r(#)=H(#, w, y1 , y2)

#=( y1+y2) f (#)&b,

with

v f (#)=(exp(a#)&1)�#;

v a=log(w+y1)(w+y2) w&1(w+y1+y2)&1;

v b=y1 y2(w( y1+y2)+2y1 y2) w&1(w+y1)&1 (w+y2)&1.

It is easily proved that f is increasing and that

r(1)=&( y1 y2)2 (w+(w+y1+y2))

w(w+y1)(w+y2)(w+y1+y2)



and

r(&1)=&2( y1 y2)2

w(w+y1)(w+y2)�0.

Suppose that 0<#�1. If y1+y2 is positive, then r is increasing in #.Thus, r(#)�r(1)�0. If y1+y2 is negative, r is decreasing in #. Thus,r(#)�r(&1)�0. This proves that H is nonpositive when # is less than 1.

If # is larger than 1, H is positive for some values of (w, y1 , y2). Indeed,fix y1 , y2 positive and let w approach 0. Then the term with w# in thedenominator dominates and makes H positive. K

REFERENCES

1. D. E. Bell, The value of pre-decision side bets for utility maximizers, Manage. Sci. 34(1988), 797�800.

2. J. Caballe� and A. Pomansky, Mixed risk aversion, J. Econ. Theory (1996), forthcoming.3. D. Duffie and R. Rahi, Financial market innovation and security design: An introduction,

J. Econ. Theory 65 (1995), 1�42.4. L. Eeckhoudt, C. Gollier, and M. Levasseur, The economics of adding and subdividing

independent risks: some comparative statics results, J. Risk Uncertainty 8 (1994), 325�337.5. C. Gollier, Repeated optional gambles and risk aversion, Manage. Scci. (1996), forth-

coming.6. C. Gollier and M. S. Kimball, Toward a systematic approach to the economic effects of

risk II: characterizing utility functions, unpublished manuscript, University of Toulouse,1994.

7. C. Gollier and R. J. Zeckhauser, Time horizon length and risk-aversion, unpublishedpaper, University of Toulouse, France, 1995.

8. M. Jackson and J. Peck, Speculation and price fluctuations with private, extrinsic signals,J. Econ. Theory 55 (1991), 247�295.

9. J. W. Pratt, Multiattribute utility and derived utility, in ``Towards Interactive andIntelligent Decision Support Systems. Proceedings of the Seventh International Con-ference on Multiple Criteria Decision Making, Kyoto'' (Y. Sawaragi, K. Inoue, andH. Nakayama, Eds.), pp. 149�154, 1987.

10. J. W. Pratt and R. Zeckhauser, Proper risk aversion, Econometrica 55 (1987), 143�154.11. M. Rothschild and J. E. Stiglitz, Increasing Risk II: Its Economic Consequences, J. Econ.

Theory 3 (1971), 225�243.12. K. Shell, Speculative equilibria, Monnaie et allocation intertemporelle (title and abstract

in French, text in English), Se� minaire d'Econome� trie Roy�Malinvaud, Centre National dela Recherche Scientifique, Paris, Nov. 21, 1977.


Investment Flexibility and the Acceptance of Risk

Documents

Transcript of Investment Flexibility and the Acceptance of Risk